Does this relationship hold true for every positive real number n?
{√(1)}+{√(2)}+{√(3)}+...+{√(n^{2})} ≤ ((n^{2}1)/12)
• If not, provide an example.
• If so, prove it.
Note: ⌊x⌋ is floor of x, which is the greatest integer less than or equal to x, and: {x} = x⌊x⌋.
https://www.desmos.com/calculator/gfedi1mdgx
A graph shows the bound itself (in blue) is pretty weak. I added what appears to be a stronger bound (in purple.)
The function itself shows some interesting structure if you zoom out.

Posted by Jer
on 20230917 11:34:43 